Periodic Solutions in the Generalized Sitnikov $(N+1)$-Body Problem

Abstract

This paper studies a special restricted (N + 1)-body problem which can be reduced to the Sitnikov problem with an appropriate positive parameter. According to the number of bodies we prove the existence (or nonexistence) of a finite (or infinite) number of symmetric families of periodic solutions. These solutions bifurcate from the equilibrium at the center of mass of the system. Introduction.In celestial mechanics there is a restricted 3-body problem known as the Sitnikov problem. In this problem we have two material points (bodies) P 1 , P 2 (called primaries) with equal mass m 1 = m 2 moving in the plane x, y around the origin (center of mass) describing elliptic orbits of the 2-body problem and a body P 0 of infinitesimal mass that moves along the z-axis. The Sitnikov problem deals with the study of the orbits of P 0 . If we choose m 1 = m 2 = 1/2, the gravitational constant G = 1, and the period of the orbit described by the primaries is equal to 2π, then the equation of motion of P 0 is

A separating surface for Sitnikov-liken+1-body problems

“…Similar work was done independently by Marchesin and Castihlo in [16]. Existence results were extended to a generalized Sitnikov problem, which involves more than two masses in a planar configuration whose orbits are ellispses, by Rivera in 2013 (see [25]). Interestingly, the results of Rivera's work included an upper bound of 234 masses in the planar configuration needed for the results to hold.…”

Quantitative Stability of Certain Families of Periodic Solutions in the Sitnikov Problem

“…The estimation of the upper bound r 0 for the canonical solutions deserves a comment; it does not depend on the branch and has to be valid for the whole [0, ξ * ] interval of initial conditions. We have computed R 0 (ξ) (equation (21)) with 0 ≤ ξ ≤ ξ * (see figure 3) and its supreme value r 0 . In tables 2 and 3 we list the coefficients that do depend on the specific branch p for N= 1 and N = 3 respectively:Ê, ∆ (0), K and µ 0 .…”

Periodic oscillations in the restricted Hip-Hop 2N+1 body problem

“…Concerning the existence of periodic motions for the plus one body in the Sitnikov problem, we refer to [10,12] where the authors prove the existence of families of symmetric periodic orbits which depend continuously on the eccentricity of the primaries, by means of the Leray-Schauder continuation method. Using the same technique, in [15] the author proves that these families also exist in a particular generalization of the Sitnikov problem where the number of primaries is n ≥ 2. More precisely, in [15] the authors find the existence of periodic motions in a restricted (n + 1)-body problem where each primary body is at the vertex of a regular n-gon and moves on elliptic orbits lying in the xy-plane around their barycenter and the plus one body moves on the z-axis.…”